Questions tagged [bivariate-distributions]

For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

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1answer
39 views

Finding conditional expectation $E(Y|X>x)$ of Bivariate Normal distribution

$(X,Y)$ follows bivariate normal distribution with parameters $(\mu_1, \mu_2, \sigma_1^2, \sigma_2^2, \rho)$. Find the conditional expectation $E(Y|X>x)$. I know that $Y|x$ follows Normal ...
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2answers
33 views

Find $\rho \left( {XY,Y} \right)$ given that $(X,Y)$ follows a bivariate normal distribution

Suppose $X$ and $Y$ have a bivariate normal distribution with parameters ${\mu _X} = {\mu _Y} = 0$, ${\sigma _X}^2 = {\sigma _Y}^2 = 1$, and $\rho = {\rho _{X,Y}} \ne 0$. I'm asked to find the ...
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2answers
45 views

The $F(b,d)-F(b,c)-F(a,d)+F(a,c)\geqslant 0$ condition of joint CDFs

I'm trying to verify that a certain function of two variables $F(x,y)$ satisfies the conditions of a joint CDF. Showing that each condition holds has been fairly straightforward except, that is, for ...
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1answer
23 views

Marginal pdf with nonnegative random variables

Suppose that $X_1$ and $X_2$ are two nonnegative random variables that satisfy $$P(X_1 > x_1, X_2 > x_2) = \exp \left[−\mu x_1 − \nu x_2 − \lambda \max(x_1, x_2)\right] ,\text{ for }x_1 \ge 0 \...
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1answer
69 views

Conditional expectation of $X$ given $X+Y=5$ of a bivariate normal distribution

Random variables X and Y have a bivariate normal distribution. If the parameters are $\sigma_x,\sigma_y,\mu_x, \mu_y, \rho$, how do we express $E(X|X+Y=5)$ using those parameters? The conditional ...
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1answer
54 views

Having trouble grasping bivariate probability distributions

So I'm having trouble grasping bivariate distribution functions. I seem to be struggling with how to determine the upper and lower bounds of my integrals. So for example, this is a question from my ...
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0answers
41 views

Joint distribution of two brownian bridges

Consider $X_t=B_t-tB_1$ for $0\leq t\leq 1$, where $B_t$ represents the standard Brownian motion. I derived that $E(X_t)=0$ and $\operatorname{Cov}(X_t,X_s)=\min(t,s)-st$. Now, I am asked the joint ...
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1answer
34 views

Generate bivariate random numbers from a joint probability distribution in Python

I have two distributions over two parameters h and t. h is Weibull distributed while t is conditioned on h and it is log-normal distributed: ...
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47 views

How to find the conditional expectation of dependent random variables?

I am wondering how to find an expression for $E[X | Y \geq a]$ for $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma_Y^2)$ and a given constant $a \in R$ in the case that $X$ and $Y$ are not ...
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15 views

Covariance in Robot Pose from Covariance in Line Segments

The solution to this might be completely obvious, so I apologize beforehand. The purpose of this question is for me to get better intuition on how to use covariance matrices of correlated variables in ...
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1answer
16 views

Showing that two random variables are standard normal but are not bivariate normal

Question: If $X\sim \mathcal{N}(0,1)$ and we define $Y$ such that $$ Y = \begin{cases} X& \text{ if }|X|<a \\ -X& \text{ if }|X|\geq a. \end{cases} $$ Show that $Y\sim \mathcal{N}(0,1)$ ...
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1answer
40 views

$X$ and $Y$ have a joint normal distribution with unity variance, zero mean, and correlation = $0.5$ — what is $P(X > 2Y | X > 0)$?

I know how to find $P(X > 2Y | X > 0)$ for the case where $X$ and $Y$ are independent. I would use a graphical approach. It would come out to be $\frac{\frac{\pi}{2} - \arctan(0.5)}{\pi}$. With ...
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1answer
61 views

$P(X > cY)$ where $X,Y$ are gaussian distributed

I believe that as long as $X,Y$ have zero mean, then regardless of what their variance is, and for any constant $c$, $P(X > cY)$ should always be $0.5$. It's easier to visualize if you think about ...
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1answer
24 views

How can you prove that a conditional bivariate Gaussian is a univariate Gaussian?

Is there a way to prove that a bivariate Gaussian becomes a univariate Gaussian when conditioned on one of the two variables?
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1answer
18 views

Variance for grouped bivariate data.

If for example I have two random variables X and Y. X stands for height and Y for weight. I compute the conditional variance for Y at each X. ( Both are discrete). Then can I compute the total ...
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28 views

Expected value of sum of square of components of a bivariate normal distribution (solved)

Suppose (X, Y) follows a bivariate normal distribution $\left(\begin{matrix}X\\Y\end{matrix}\right)\sim N \left(\begin{matrix}\left(\begin{matrix}0\\0\end{matrix}\right),\left(\begin{matrix}1&\...
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30 views

How to determine distribution of $s_1^2/s_2^2$

Let $x_i, y_i , i=1,2,...,n$ be a random sample from a bi-variate normal population $N_2(μ_1,μ_2,σ_1,σ_2,ρ)$. How to find the distribution of $s_1^2/s_2^2$ in non-null case i.e. when $ρ≠0$
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65 views

Gaussian and Circles/ Ellipse

Suppose you have a standard Gaussian vector $Z=(Z_1,Z_2)$. I want to find $\Bbb P(Z_1>0$ and $0<Z_2<\sqrt3Z_1$). I did this by first drawing a circle, of radius $1$, representing the Standard ...
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1answer
50 views

What is the PDF of the sum of two dependent variables?

Let $X \sim \mathcal{N}(\mu_1, \sigma_1)$ and $Y \sim \mathcal{N}(\mu_2, \sigma_2)$ denote two independent normal random variables, then what is the PDF of $Z=(X+Y)^2-X^2=2XY+Y^2$? The problem is hard ...
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14 views

Find the Pearson correlation coefficient

Question I tried solving this question using bivariate frequency distribution but it seems the answer(My Ans = 0.404 ) I've got is incorrect (Correct Ans = 0.6241). I double checked my calculations ...
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1answer
66 views

Given $(X,Y)$ bivariate normal, find a linear combination independent of $Y$ [closed]

I have a question about the answer to this question. At the start of the proof we assume that $$\frac{X-\mu_1}{\sigma_1} = \rho \frac{Y-\mu_2}{\sigma_2} + \sqrt{1-\rho^2} Z \tag{$*$}$$ But I can't ...
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21 views

Density of Bivariate Gaussian distribution in Kalman Filter

Given $$\textbf{p}_{t+1}=\textbf{A}\textbf{p}_t+\textbf{n}_{t+1} $$ How can I verify $f(\textbf{p}_1)$ is the density of a bivariate gaussian distribution, given $$f(\textbf{p}_1)=\int f(\textbf{p}_1|\...
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35 views

bivariate conditional Gaussian

I want to calculate this probability in terms of $Q$-functions. $Pr\{X>m , Y>m , Y>X\}$ where $Y\sim N(0,\sigma_y^2)$ and $X\sim N(\mu,\sigma_x^2)$ and m is constant. Also $X$ and $Y$ are ...
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0answers
37 views

Bivariate normal distribution, the 5th parameter

What does $\rho$ , the 5th parameter mean in the bivariate normal distribution ?
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2answers
69 views

Probability that $X < Y$ for bivariate normal variable

For the following probability questions, I'm a bit lost on how to proceed. Let $X,Y$ be bivariate normal random variables with $\mathbb{E}(X)=10$, $\mathbb{E}(Y)=-7$, $Var(X)=30$, $Var(Y)=100$, and $\...
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1answer
136 views

Derivation of bivariate Gaussian copula density

The multivariate Gaussian copula density, derived here, is $$c(u_1,\ldots,u_n;\Sigma)=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}(\Sigma^{-1}-I)x\right)$$ where $\Sigma$ is the covariance ...
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1answer
47 views

Bivariate Normal Transformation

I have $X, Y$ governed by a Bivariate Normal($\mu_{X} = 0, \mu_{y} = 0, \sigma^2_{x} = 1, \sigma^2_{y} = 1, \rho=\rho$). $H, J$ are from the same joint distribution as $X,Y$ and are independent. If $G ...
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1answer
47 views

Let $f(x,y,z)=e^{-x-y-z},x>0,y>0,z>0$, and $=0$ otherwise, be the joint PDF of $(X,Y,Z)$. Compute $P(X=Y<Z)$. [closed]

Let $$f(x,y,z)=\begin{cases}e^{-x-y-z}, \ \ \text{$x>0,y>0,z>0$} \\0,\ \ \text{otherwise}\end{cases}$$ be the joint PDF of $(X,Y,Z)$. Compute $P(X=Y<Z)$. According to the textbook, the ...
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1answer
100 views

How to calculate the expected value of bivariate normal distribution?

Let $X= (X_1,X_2)$ be a random vector with bivariate normal distribution $X\sim N(\mu,\Sigma)$ such that $X_1$ and $X_2$ are positively correlated and we also have to: $P(X_1<1) = 0,84134 $, $P(...
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1answer
31 views

How do you find Pr$(X>Y)$ in bivariate normal distribution given that $X$ and $Y$ are independent?

Specifically, if company A's sales of widgets for the upcoming year are normally distributed with mean 10,000 and standard deviation 2,000, while company B's sales of widgets for the upcoming year are ...
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1answer
76 views

Finding expectation of minimum of $(X,Y)$ where $(X,Y)$ is bivariate normal distribution.

Let $(X,Z)$ be bivariate normal with parameters $\mu_X := E(X) = 1, \mu_Z := E(Z ) = 1, \sigma_X^2 := Var(X) = 1$, $ \sigma_Z^2 := Var(Z ) = 1$, and the correlation coefficient of (X, Z ) is $\rho$ ...
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0answers
27 views

Show the following integral for non-negative X and Y

Completely stuck on this problem. Let X and Y be non-negative random variables with an arbitrary joint probability distribution function. Let $$I(x,y)=\begin{cases}1,\quad if \:X>x,\:Y>y\\0,\...
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1answer
42 views

Given two indep standard normal RVs, X and Y, what is $P(X^2 + Y^2 \leq 1)$?

Given two standard normal random RVs, X and Y, how do you find $P(X^2 + Y^2 \leq 1)$? I approached this by integrating the bivariate using polar coordinates as $x^2 + y^2 = 1$. Not sure if this is ...
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0answers
17 views

Finding the covariance matrix of $\hat{\theta_g}$

I am stuck in finding covariance matrix of $\hat{\theta_g}=\sum_{t=1}^{N} z_t |\Sigma|^{\frac{1}{2}} e^{\frac{1}{2} z_t' (\Sigma^{-1}-I)z_t}$ , where my $z_t=(u_t,v_t) \sim N_2(0,\Sigma)$. It is given ...
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2answers
72 views

Why is $\frac{X}{Y} = \frac{X}{\vert Y \rvert}$ for $X, Y \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(0,1)$?

The justification provided was "symmetry of the Normal", but as far as I understand it $Y$ is not equivalent to $\lvert Y \rvert$, most obviously because the supports are no longer the same! ...
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0answers
41 views

A “perturbed” correlation coefficient for bivariate normal distribution

consider a random vector $ (X_1,X_2) $ following a bivariate normal distribution with mean vector $ \mu=(\mu_1,\mu_2) $ and (symmetric, positive semi-definite) covariance matrix $ \Sigma=\begin{...
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1answer
56 views

Joint distribution table for coin flips

I am tossing a fair coin 6 times. I define $X$ to be the number of heads in first 3 tosses and $Y$ to be the number of heads in all 6 tosses. $X$ and $Y$ are random variables. I want to create the ...
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0answers
149 views

Moment generating function of the maximum of two normal random variables

I found this paper here. In equations (5) - (6), the paper says that suppose $(X_1,X_2)$ are jointly normal with means and variances $(\mu_1,\mu_2),(\sigma_1^2,\sigma_2^2)$, define $X = \max\{X_1,X_2\}...
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1answer
52 views

How to compute expectation for the product of squared jointly normal random variables

Let $X_1$ and $X_2$ be jointly normal random variables with $\mathbb E[X_1] = \mathbb E[X_2] = 0$ and $\operatorname{var}(X_1) = \sigma_1^2$, $\operatorname{var}(X_2) = \sigma_2^2$. The correlation ...
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3answers
212 views

Distribution of $Z=\frac{X}{X+Y}$ when $X,Y$ are i.i.d geometric random variables

If $X,Y$ are independent and have same geometric distribution with parameter $p$, find the distribution of $Z=\frac{X}{X+Y}$. The solution is $$\mathbb{P}\left(Z=\frac{m}{n}\right)=\sum_k\mathbb{P}(X=...
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2answers
52 views

What are $c$'s in antiderivative of double integral in Wolfram Alpha

I would like to solve the following double integral: $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x y)^{-\theta - 1} dx \hspace{1mm} dy$$ Using Wolfram Alpha for symbolic algebra, it gives a ...
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2answers
80 views

Uniform Bivariate Probabilities

Suppose I have a uniformly distributed variable X on the interval $[0,a]$ and a uniformly distributed variable Y on the interval $[0,b]$. Thus we have the respected pdfs as $f(x)=\frac{1}{a}$ and $f(y)...
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2answers
110 views

If $(X,Y)$ is bivariate normal with correlation $\rho$ and $\sigma_X^2 = \sigma_Y^2$, show that $X$ and $Y - \rho X$ are independent

Let $(X,Y)$ be bivariate normal with correlation $\rho$ and variance $\sigma_X^2=\sigma_Y^2$. Show that $X$ and $Y - \rho X$ are independent. I found there's a general result which states that if $(X, ...
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1answer
125 views

Bivariate normal MLE confidence interval question

Consider a random sample $(L_1, R_1), . . . ,(L_n, R_n)$ of eye pressure measurements in the left and right eyes of n patients. a) Suppose that $L_j$ and $R_j$ are independent and normally distributed ...
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2answers
98 views

Sufficient condition for Independence of two random variables

$X$, $W$, and $Y=\frac{X}{\sigma_x}-\frac{ρ(x, w)*W}{\sigma_w}$ are normal random variables with probability density functions $f(x)$, $f(w)$ and $f(y)$. $X$ and $W$ are bivariate normal, $f(x) * f(...
2
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1answer
60 views

Distribution after adding noise to a Gaussian Distribution

I have a random variable $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$. Now, I add a noise $N \sim \mathcal{N}(0, \sigma_N^2)$ to $X$ to get $Y$ ($Y = X + N$). Thus, $Y \sim \mathcal{N}(\mu_X, \sigma_X^2 + \...
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0answers
19 views

Using log-concavity to show variance is log-concave?

Suppose I have a bivariate distribution $X,Y$ on $\mathbb{R}^2$, which is log-concave. Say it has pdf $\zeta(x,y)$. Also, let $A = (1-k)X + kY$ $B = kX - (1-k)Y$ I'd like to prove that $E_A[Var(B|A)]$ ...
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1answer
47 views

Evaluating $ \mathbb{P}(Y+X>a \cap X>b)$ when $(X,Y)$ is bivariate normal

This is a rather simple question: we have $X,Y$ jointly Bivariate Normally distributed, with $f_{X,Y}(x,y)$ being their density. We're interested in the probabilities of the type: $$ \mathbb{P}(Y+X>...
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1answer
65 views

Find $\theta$ such that $W = X \cosθ +Y \sinθ \text{ and } Z = X \cosθ −Y \sinθ$ are independent.

Find $\theta$ such that $W = X \cosθ +Y \sinθ \text{ and } Z = X \cosθ −Y \sinθ$ are independent. It is given that X and Y be jointly normal each with mean $0$ and variance $1$. I have shown that $$W \...

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